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Abstract

This paper deals with the application of temporal averaging methods to recurrent networks
of noisy neurons undergoing a slow and unsupervised modification of their connectivity
matrix called learning. Three time-scales arise for these models: (i) the fast neuronal
dynamics, (ii) the intermediate external input to the system, and (iii) the slow learning
mechanisms. Based on this time-scale separation, we apply an extension of the mathematical
theory of stochastic averaging with periodic forcing in order to derive a reduced
deterministic model for the connectivity dynamics. We focus on a class of models where
the activity is linear to understand the specificity of several learning rules (Hebbian,
trace or anti-symmetric learning). In a weakly connected regime, we study the equilibrium
connectivity which gathers the entire ‘knowledge’ of the network about the inputs.
We develop an asymptotic method to approximate this equilibrium. We show that the
symmetric part of the connectivity post-learning encodes the correlation structure
of the inputs, whereas the anti-symmetric part corresponds to the cross correlation
between the inputs and their time derivative. Moreover, the time-scales ratio appears
as an important parameter revealing temporal correlations.

Keywords:

1 Introduction

Complex systems are made of a large number of interacting elements leading to non-trivial
behaviors. They arise in various areas of research such as biology, social sciences,
physics or communication networks. In particular in neuroscience, the nervous system
is composed of billions of interconnected neurons interacting with their environment.
Two specific features of this class of complex systems are that (i) external inputs
and (ii) internal sources of random fluctuations influence their dynamics. Their theoretical
understanding is a great challenge and involves high-dimensional non-linear mathematical
models integrating non-autonomous and stochastic perturbations.

Modeling these systems gives rise to many different scales both in space and in time.
In particular, learning processes in the brain involve three time-scales: from neuronal
activity (fast), external stimulation (intermediate) to synaptic plasticity (slow).
Here, fast time-scale corresponds to a few milliseconds and slow time-scale to minutes/hour,
and intermediate time-scale generally ranges between fast and slow scales, although
some stimuli may be faster than neuronal activity time-scale (e.g., submilliseconds auditory signals [1]). The separation of these time-scales is an important and useful property in their
study. Indeed, multiscale methods appear particularly relevant to handle and simplify
such complex systems.

First, stochastic averaging principle [2,3] is a powerful tool to analyze the impact of noise on slow-fast dynamical systems.
This method relies on approximating the fast dynamics by its quasi-stationary measure
and averaging the slow evolution with respect to this measure. In the asymptotic regime
of perfect time-scale separation, this leads to a slow reduced system whose analysis
enables a better understanding of the original stochastic model.

Second, periodic averaging theory [4], which has been originally developed for celestial mechanics, is particularly relevant
to study the effect of fast deterministic and periodic perturbations (external input)
on dynamical systems. This method also leads to a reduced model where the external
perturbation is time-averaged.

It seems appropriate to gather these two methods to address our case of a noisy and
input-driven slow-fast dynamical system. This combined approach provides a novel way
to understand the interactions between the three time-scales relevant in our models.
More precisely, we will consider the following class of multiscale stochastic differential
equations (SDEs), with two small parameters

(1)

where represents the fast activity of the individual elements, represents the connectivity weights that vary slowly due to plasticity, and represents the value of the external input at time t. Random perturbations are included in the form of a diffusion term, and is a standard Brownian motion.

We are interested in the double limit and to describe the evolution of the slow variable w in the asymptotic regime where both the variable v and the external input are much faster than w. This asymptotic regime corresponds to the study of a neuronal network in which both
the external input u and the neuronal activity v operate on a faster time-scale than the slow plasticity-driven evolution of synaptic
weights W. To account for the possible difference of time-scales between v and the input, we introduce the time-scale ratio . In the interesting case where , one needs to understand the long-time behavior of the rescaled periodically forced
SDE for any fixed

Recently, in an important contribution [5], a precise understanding of the long-time behavior of such processes has been obtained
using methods from partial differential equations. In particular, conditions ensuring
the existence of a periodic family of probability measures to which the law of v converges as time grows have been identified, together with a sharp estimation of
the speed of mixing. These results are at the heart of the extension of the classical
stochastic averaging principle [2] to the case of periodically forced slow-fast SDEs [6]. As a result, we obtain a reduced equation describing the slow evolution of variable
w in the form of an ordinary differential equation,

where is constructed as an average of G with respect to a specific probability measure, as explained in Section 2.

This paper first introduces the appropriate mathematical framework and then focuses
on applying these multiscale methods to learning neural networks.

The individual elements of these networks are neurons or populations of neurons.
A common assumption at the basis of mathematical neuroscience [7] is to model their behavior by a stochastic differential equation which is made of
four different contributions: (i) an intrinsic dynamics term, (ii) a communication
term, (iii) a term for the external input, and (iv) a stochastic term for the intrinsic
variability. Assuming that their activity is represented by the fast variable , the first equation of system (1) is a generic representation of a neural network
(function F corresponds to the first three terms contributing to the dynamics). In the literature,
the level of non-linearity of the function F ranges from a linear (or almost-linear) system to spiking neuron dynamics [8], yet the structure of the system is universal.

These neurons are interconnected through a connectivity matrix which represents the
strength of the synapses connecting the real neurons together. The slow modification
of the connectivity between the neurons is commonly thought to be the essence of learning.
Unsupervised learning rules update the connectivity exclusively based on the value
of the activity variable. Therefore, this mechanism is represented by the slow equation
above, where is the connectivity matrix and G is the learning rule. Probably the most famous of these rules is the Hebbian learning
rule introduced in [9]. It says that if both neurons A and B are active at the same time, then the synapses
from A to B and B to A should be strengthened proportionally to the product of the
activity of A and B. There are many different variations of this correlation-based
principle which can be found in [10,11]. Another recent, unsupervised, biologically motivated learning rule is the spike-timing-dependent
plasticity (STDP) reviewed in [12]. It is similar to Hebbian learning except that it focuses on causation instead of
correlation and that it occurs on a faster time-scale. Both of these types of rule
correspond to G being quadratic in v.

Previous literature about dynamic learning networks is thick, yet we take a significantly
different approach to understand the problem. An historical focus was the understanding
of feedforward deterministic networks [13-15]. Another approach consisted in precomputing the connectivity of a recurrent network
according to the principles underlying the Hebbian rule [16]. Actually, most of current research in the field is focused on STDP and is based
on the precise times of the spikes, making them explicit in computations [17-20]. Our approach is different from the others regarding at least one of the following
points: (i) we consider recurrent networks, (ii) we study the evolution of the coupled
system activity/connectivity, and (iii) we consider bounded dynamical systems for
the activity without asking them to be spiking. Besides, our approach is a rigorous
mathematical analysis in a field where most results rely heavily on heuristic arguments
and numerical simulations. To our knowledge, this is the first time such models expressed
in a slow-fast SDE formalism are analyzed using temporal averaging principles.

The purpose of this application is to understand what the network learns from the
exposition to time-dependent inputs. In other words, we are interested in the evolution
of the connectivity variable, which evolves on a slow time-scale, under the influence
of the external input and some noise added on the fast variable. More precisely, we
intend to explicitly compute the equilibrium connectivities of such systems. This
final matrix corresponds to the knowledge the network has extracted from the inputs.
Although the derivation of the results is mathematically tough for untrained readers,
we have tried to extract widely understandable conclusions from our mathematical results
and we believe this paper brings novel elements to the debate about the role and mechanisms
of learning in large scale networks.

Although the averaging method is a generic principle, we have made significant assumptions
to keep the analysis of the averaged system mathematically tractable. In particular,
we will assume that the activity evolves according to a linear stochastic differential
equation. This is not very realistic when modeling individual neurons, but it seems
more reasonable to model populations of neurons; see Chapter 11 of [7].

The paper is organized as follows. Section 2 is devoted to introducing the temporal
averaging theory. Theorem 2.2 is the main result of this section. It provides the
technical tool to tackle learning neural networks. Section 3 corresponds to application
of the mathematical tools developed in the previous section onto the models of learning
neural networks. A generic model is described and three different particular models
of increasing complexity are analyzed. First, Hebbian learning, then trace-learning,
and finally STDP learning are analyzed for linear activities. Finally, Section 4 is
a discussion of the consequences of the previous results from the viewpoint of their
biological interpretation.

2 Averaging principles: theory

In this section, we present multiscale theoretical results concerning stochastic averaging
of periodically forced SDEs (Section 2.3). These results combine ideas from singular
perturbations, classical periodic averaging and stochastic averaging principles. Therefore,
we recall briefly, in Sections 2.1 and 2.2, several basic features of these principles,
providing several examples that are closely related to the application developed in
Section 3.

2.1 Periodic averaging principle

We present here an example of a slow-fast ordinary differential equation perturbed
by a fast external periodic input. We have chosen this example since it readily illustrates
many ideas that will be developed in the following sections. In particular, this example
shows how the ratio between the time-scale separation of the system and the time-scale
of the input appears as a new crucial parameter.

Example 2.1 Consider the following linear time-inhomogeneous dynamical system with two parameters:

This system is particularly handy since one can solve analytically the first ordinary
differential equation, that is,

where we have introduced the time-scales ratio

In this system, one can distinguish various asymptotic regimes when and are small according to the asymptotic value of μ:

• Regime 1: Slow input :

First, if and is fixed, then is close to , and from geometric singular perturbation theory[21,22] one can approximate the slow variable by the solution of

Now taking the limit and applying the classical averaging principle[4] for periodically driven differential equations, one can approximate by the solution of

since .

• Regime 2: Fast input :

If and is fixed, then the classical averaging principle implies that is close to the solution of

so that can be approximated by

and when , one does not recover the same asymptotic behavior as in Regime 1.

• Regime 3: Time-scales matching :

Now consider the intermediate case where is asymptotically proportional to . In this case, can be approximated on the fast time-scale by the periodic solution of . As a consequence, will be close to the solution of

since .

Thus, we have seen in this example that

1. the two limits and do not commute,

2. the ratio μ between the internal time-scale separation and the input time-scale is a key parameter in the study of slow-fast systems subject to a time-dependent
perturbation.

2.2 Stochastic averaging principle

Time-scales separation is a key property to investigate the dynamical behavior of
non-linear multiscale systems, with techniques ranging from averaging principles to
geometric singular perturbation theory. This property appears to be also crucial to
understanding the impact of noise. Instead of carrying a small noise analysis, a multiscale
approach based on the stochastic averaging principle[2] can be a powerful tool to unravel subtle interplays between noise properties and
non-linearities. More precisely, consider a system of SDEs in :

with initial conditions , , and where is called the slow variable, is the fast variable, with F, G, Σ smooth functions ensuring the existence and uniqueness for the solution , and a p-dimensional standard Brownian motion, defined on a filtered probability space . Time-scale separation in encoded in the small parameter ϵ, which denotes in this section a single positive real number.

In order to approximate the behavior of for small ϵ, the idea is to average out the equation for the slow variable with respect to the
stationary distribution of the fast one. More precisely, one first assumes that for
each fixed, the frozen fast SDE,

and w the solution of with the initial condition . Under some dissipativity assumptions, the stochastic averaging principle [2] states:

Theorem 2.1For anyand,

(3)

As a consequence, analyzing the behavior of the deterministic solution w can help to understand useful features of the stochastic process .

Example 2.2 In this example we consider a similar system as in Example 2.1, but with a noise
term instead of the periodic perturbation. Namely, we consider the solution of the system of SDEs,

with a small parameter and a positive constant. From Theorem 2.1, the stochastic slow variable can be approximated in the sense of (3) by the deterministic solution w of

where is the stationary measure of the linear diffusion process,

that is,

Consequently, can be approximated in the limit by the solution of

Applying (3) leads to the following result: for any and ,

Interestingly, the asymptotic behavior of for small ϵ is characterized by a deterministic trajectory that depends on the strength σ of the noise applied to the system. Thus, the stochastic averaging principle appears
particularly interesting when unraveling the impact of noise strength on slow-fast
systems.

Many other results have been developed since, extending the set-up to the case where
the slow variable has a diffusion component or to infinite-dimensional settings for
instance, and also refining the convergence study, providing homogenization results concerning the limit of or establishing large deviation principles (see [23] for a recent monograph). However, fewer results are available in the case of non-homogeneous
SDEs, that is, when the system is perturbed by an external time-dependent signal.
This setting is of particular interest in the framework of stochastic learning models,
and we present the main relevant mathematical results in the following section.

with a τ-periodic function and . Parameter represents the internal time-scale separation and the input time-scale. We consider the case where both and are small, that is, a strong time-scale separation between the fast variable and the slow one , and a fast periodic modulation of the fast drift .

We further denote .

Definition 2.1 We define the asymptotic time-scale ratio

(5)

Accordingly, we denote the distinguished limit when , with .

The following assumption is made to ensure existence and uniqueness of a strong solution
to system (4). In the following, will denote the usual scalar product for vectors.

Assumption 2.1 Existence and uniqueness of a strong solution

(i) The functions F, G, and Σ are locally Lipschitz continuous in the space variable z. More precisely, for any , there exists a constant such that

(ii) There exists a constant such that

To control the asymptotic behavior of the fast variable, one further assumes the following.

Assumption 2.2 Asymptotic behavior of the fast process:

(i) The diffusion matrix Σ is bounded

and uniformly non-degenerate

(ii) There exists such that for all and for all ,

According to the value of , the stochastic averaging principle is based on a description of the asymptotic behavior
of various rescaled fast frozen processes. More precisely, under Assumptions 2.1 and 2.2,
one can deduce that:

• For any fixed and fixed, the law of the rescaled time-homogeneous frozen process,

According to the value of μ, we introduce a vector field which will play a role similar to introduced in equation (2).

Definition 2.2 We define as follows. In the time-scale matching case, that is, when , then

(7)

Notation We may denote the periodic system of measures associated with (6) by to emphasize its relationship with F and Σ. Accordingly, we may denote by .

We are now able to present our main mathematical result. Extending Theorem 2.1, the
following theorem describes the asymptotic behavior of the slow variable when with . We refer to [6] for more details about the full mathematical proof of this result.

Theorem 2.2Let. Ifwis the solution of

(8)

then the following convergence result holds, for alland:

Remark 2.1

1. The extremal cases and are not covered in full rigor by Theorem 2.2. However, the study of the sequential
limits followed by or followed by can be deduced from an appropriate combination of classical periodic and stochastic
averaging theorems:

• Slow input: If one considers the case where the limit is taken first, so that from Theorem 2.1 with fast variable and slow variables and t (with the trivial equation ), is close in probability on finite time-intervals to the solution of the following
inhomogeneous ordinary differential equation:

Then taking the limit , one can apply the deterministic averaging principle to the fast periodic vector
field , so that converges when to the solution of

where

• Fast input: If the limit is taken first, one first has to perform a classical averaging of the periodic drift
leading to the homogeneous system of SDEs (4), but with instead of . Then, an application of Theorem 2.1 on this system gives an averaged vector field

2. To study the extremal cases and in full generality, one would need to consider all the possible relationships between
and , not only the linear one as in the present article, but also of the type for example. In this case, we believe that the regime converges to the same limit as taking the limit first and the regime corresponds to taking the limit first. The intermediate regime seems to be the only one for which the limit cannot be obtained by combining classical
averaging principles. Therefore, the present article is focused on this case, in which
the averaged system depends explicitly on the scaling parameter μ. Moreover, in terms of applications, this parameter can have a relatively easy interpretation
in terms of the ratio of time-scales between intrinsic neuronal activity and typical
stimulus time-scales in a given situation. Although the zeroth order limit (i.e., the averaged system) seems to depend only on the position of α with respect to 1, it seems reasonable to expect that the fluctuations around the
limit would depend on the precise value of α. This is a difficult question which may deserve further analysis.

The case is already very rich in the sense that it combines simultaneously both the periodic
and stochastic averaging principles in a new way that cannot be recovered by sequential
applications of those principles. A particular role is played by the frozen periodically-forced
SDE (6). The equivalent of the quasi-stationary measure of Theorem 2.1 is given by the asymptotically periodic behavior of equation (6),
represented by the periodic family of measures .

3. By a rescaling of the frozen process (6), one deduces the following scaling relationships:

and

Therefore, if one knows, in the case , the averaged vector field associated with the fast process generated by a drift
F and a diffusion coefficient σ, denoted , it is possible to deduce in the general case with a change and .

4. It seems reasonable to expect that the above result is still valid when considering
ergodic, but not necessarily periodic, time dependency of the function . In equation (7), instead of integrating over one period, one should integrate it with respect to an ergodic stationary measure.
However, this extension requires non-trivial technical improvements of [5] which are beyond the scope of this paper.

2.3.1 Case of a fast linear SDE with periodic input

We present here an elementary case where one can compute explicitly the quasi-stationary
time-periodic family of measures , when the equation for the fast variable is linear. Namely, we consider the solution of

with initial condition , and where is a matrix whose eigenvalues have positive real parts and is a τ-periodic function.

We are interested in the large time behavior of the law of , which is a time-inhomogeneous Ornstein-Uhlenbeck process. From [5] we know that its law converges to a τ-periodic family of probability measures . Due to the linearity in the previous equation, is Gaussian with a time-dependent mean and a constant covariance matrix

where is the -periodic attractor of , i.e.,

and Q is the unique solution of the Lyapunov equation

(9)

Indeed, if one denotes , then is a solution of the classical homogeneous Ornstein-Uhlenbeck equation

whose stationary distribution is known to be a centered Gaussian measure with the
covariance matrix Q solution of (9); see Chapter 3.2 of [24]. Notice that if A is self-adjoint with respect to (i.e., ), then the solution is , which will be used in Appendix B.2.

Hence, in the linear case, the averaged vector field of equation (7) becomes

(10)

where is the probability density function of the Gaussian law with mean and covariance .

Therefore, due to the linearity of the fast SDE, the periodic system of measure ν is just a constant Gaussian distribution shifted by a periodic function of time . In case G is quadratic in v, this remark implies that one can perform independently the integral over time and
over in formula (10) (noting that the crossed term has a zero average). In this case,
contributions from the periodic input and from noise appear in the averaged vector
field in an additive way.

Example 2.3 In this last example, we consider a combination between Example 2.1 and Example 2.2,
namely we consider the following system of periodically forced SDEs:

As in Example 2.1 and as shown above, the behavior of this system when both and are small depends on the parameter μ defined in (5). More precisely, we have the following three regimes:

• Regime 1: slow input:

• Regime 2: fast input:

• Regime 3: time-scale matching:

2.4 Truncation and asymptotic well-posedness

In some cases, Assumptions 2.1-2.2 may not be satisfied on the entire phase space
, but only on a subset. Such situations will appear in Section 3 when considering
learning models. We introduce here a more refined set of assumptions ensuring that
Theorem 2.2 still applies.

Let us start with an example, namely the following bi-dimensional system with white
noise input:

(11)

with , , , .

For the fast drift to be non-explosive, it is necessary to have with for all time. The concern about this system comes from the fact that the slow variable
w may reach l due to the fluctuations captured in the term , for instance, if κ is not large enough. Such a system may have exponentially growing trajectories. However,
we claim that for small enough ϵ, will remain close to its averaged limit w for a very long time, and if this limit remains below , then can be considered as well-posed in the asymptotic limit . To make this argument more rigorous, we suggest the following definition.

Definition 2.3 A stochastic differential equation with a given initial condition is asymptotically
well posed in probability if for the given initial condition,

1. a unique solution exists until a random time

2. for all ,

We give in the following proposition sufficient conditions for system (4) to be asymptotically
well posed in probability and to satisfy conclusions of Theorem 2.2.

Let us introduce the following set of additional assumptions.

Assumption 2.3 Moment conditions:

(i) There exists such that

(ii) For any and any bounded subset K of ,

Remark 2.2 This last set of assumptions will be satisfied in all the applications of Section 3
since we consider linear models with additive noise for the equation of v, implying this variable to be Gaussian and the function G only involves quadratic moments of v; therefore, the moment conditions (i) and (ii) will be satisfied without any difficulty.
Moreover, if one considers non-linear models for the variable v, then the Gaussian property may be lost; however, adding sigmoidal non-linearity
has, in general, the effect of bounding the dynamics, thus making these moment assumptions
reasonable to check in most models of interest.

Property 2.3If there exists a subset ℰ ofsuch that

1. The functionsF, G, Σsatisfy Assumptions 2.1-2.3 restricted on.

2. ℰ is invariant under the flow of, as defined in (7).

Then for any initial condition, system (4) is asymptotically well posed in probability andsatisfies the conclusion of Theorem 2.2.

Proof See Appendix A.2. □

Here, we show that it applies to system (11). First, with , for some , it is possible to show that Assumptions 2.1-2.2 are satisfied on . Then, as a special case of (10), we obtain the following averaged system:

It remains to check that the solution of this system satisfies

that is, the subset is invariant under the flow of .

This property is satisfied as soon as

Indeed, one can show that admits two solutions iff ,

and that is stable whereas is unstable. Thus, if with , then for all . In fact, the invariance property is true for all .

3 Averaging learning neural networks

In this section, we apply the temporal averaging methods derived in Section 2 on models
of unsupervised learning neural networks. First, we design a generic learning model
and show that one can define formally an averaged system with equation (7). However,
going beyond the mere definition of the averaged system seems very difficult and we
only manage to get explicit results for simple systems where the fast activity dynamics
is linear. In the three last subsections, we push the analysis for three examples
of increasing complexity.

In the following, we always consider that the initial connectivity is 0. This is an
arbitrary choice but without consequences, because we focus on the regime where there
is a single globally stable equilibrium point (see Section 3.2.3).

3.1 A generic learning neural network

We now introduce a large class of stochastic neuronal networks with learning models.
They are defined as coupled systems describing the simultaneous evolution of the activity
of neurons and the connectivity between them. We define , the activity field of the network, and , the connectivity matrix.

Each neuron variable is assumed to follow the SDE

where the function characterizes the intrinsic non-linear dynamical behavior of neuron i and is the input received by neuron i. The stochastic term is added to account for internal sources of noise. In terms of notations, is a standard n-dimensional Brownian motion, Σ is an matrix, possibly function of v or other variables, and denotes the ith component of the vector .

The input to neuron i has mainly two components: the external input and the input coming from other neurons in the network . The latter is a priori a complex combination of post-synaptic potentials coming from many other neurons.
The coefficient of the connectivity matrix accounts for the strength of a synapse . Note that neurons can be connected to themselves, i.e., is not necessarily null. Thus, we can write

where and ℋ is a function taking the history of and and returning a real for each time t (to take convolutions into account). In practical cases, they are often taken to
be sigmoidal functions. We abusively redefine and ℋ as vector valued operators corresponding to the element-wise application of
their real counterparts. We also define the function such that . Together with a slow generic learning rule, this leads to defining a stochastic learning model as the following system of SDEs.

Definition 3.1

Before applying the general theory of Section 2, let us make several comments about
this generic model of neural network with learning. This model is a non-autonomous,
stochastic, non-linear slow-fast system.

In order to apply Theorem 2.2, one needs Assumptions 2.1, 2.2, and 2.3 to be satisfied,
restricting the space of possible functions , ℋ, ℱ, Σ, and G. In particular, Assumption 2.2(ii) seems rather restrictive since it excludes systems
with multiple equilibria and suggests that the general theory is more suited to deal
with rate-based networks. However, one should keep in mind that these assumptions
are only sufficient, and that the double averaging principle may work as well in systems
which do not satisfy readily those assumptions.

As we will show in Section 3.3, a particular form of history-dependence can be taken
into account, to a certain extent. Indeed, for instance, if the function ℱ is actually
a functional of the past trajectory of variable which can be expressed as the solution of an additional SDE, then it may be possible
to include a certain form of history-dependence. However, purely time-delayed systems
do not enter the scope of this theory, although it might be possible to derive an
analogous averaging method in this framework.

The noise term can be purely additive or set by a particular function as long as it satisfies Assumption 2.2(i), meaning that it must be uniformly non-degenerate.

In the following subsection, we apply the averaging theory to various combinations
of neuronal network models, embodied by choices of functions , ℋ, ℱ, Σ, and various learning rules, embodied by a choice of the function G. We will also analyze the obtained averaged system, describing the slow dynamics
of the connectivity matrix in the limit of perfect time-scale separation and, in particular,
study the convergence of this averaged system to an equilibrium point.

3.2 Symmetric Hebbian learning

One of the simplest, yet non-trivial, stochastic learning models is obtained when
considering

• A linear model for neuronal activity, namely with l a positive constant.

where neurons are assumed to have the same decay constant: ; u is a periodic continuous input (it replaces in the previous section); with and is n-dimensional Brownian noise.

The first question that arises is about the well-posedness of the system: What is
the definition interval of the solutions of system (12)? Do they explode in finite
time? At first sight, it seems there may be a runaway of the solution if the largest
real part among the eigenvalues of W grows bigger than l. In fact, it turns out this scenario can be avoided if the following assumption linking
the parameters of the system is satisfied.

Assumption 3.1 There exists such that

where .

It corresponds to making sure the external (i.e., ) or internal (i.e., σ) excitations are not too large compared to the decay mechanism (represented by κ and l). Note that if , and d are fixed, it is sufficient to increase κ or l for this assumption to be satisfied.

Under this assumption, the space

is invariant by the flow of the averaged system , where means W is semi-definite positive and means is definite positive. Therefore, the averaged system is defined and bounded on . The slow/fast system being asymptotically close to the averaged system, it is therefore
asymptotically well-defined in probability. This is summarized in the following theorem.

Theorem 3.1If Assumption 3.1 is verified for, then system (12) is asymptotically well posed in probability and the connectivity matrix, the solution of system (12), converges toWin the sense that for all,

whereWis the deterministic solution of

(13)

whereis the-periodic attractor of, whereis supposed to be fixed.

Proof See Theorem B.1 in Appendix B.2. □

In the following, we focus on the averaged system described by (13). Its right-hand
side is made of three terms: a linear and homogeneous decay, a correlation term, and
a noise term. The last two terms are made explicit in the following.

3.2.1 Noise term

As seen in Section 2, in the linear case, the noise term Q is the unique solution of the Lyapunov equation (9) with and . Because the noise is spatially uncorrelated and identical for each neuron and also
because the connectivity is symmetric, observe that is the unique solution of the system.

In more complicated cases, the computation of this term appears to be much more difficult
as we will see in Section 3.4.

3.2.2 Correlation term

This term corresponds to the auto-correlation of neuronal activity. It is only implicitly
defined; thus, this section is devoted to finding an explicit form depending only
on the parameters l, μ, τ, the connectivity W, and the inputs u. Actually, one can perform an expansion of this term with respect to a small parameter
corresponding to a weakly connected expansion. Most terms vanish if the connectivity W is small compared to the strength of the intrinsic decaying dynamics of neurons l.

The auto-correlation term of a -periodic function can be rewritten as

With this notation, it is simple to think of v as a ‘semi-continuous matrix’ of . Hence, the operator ‘⋅’ can be though of as a matrix multiplication. Similarly,
the transpose operator turns a matrix into a matrix . See Appendix B.1 for details about the notations.

It is common knowledge, see [17] for instance, that this term gathers information about the correlation of the inputs.
Indeed, if we assume that the input is sufficiently slow, then has enough time to converge to for all . Therefore, in the first order . This leads to . In the weakly connected regime, one can assume that leading to which is the auto-correlation of the inputs.

Actually, without the assumption of a slow input, lagged correlations of the input
appear in the averaged system. Before giving the expression of these temporal correlations,
we need to introduce some notations. First, define the convolution filter , where H is the Heaviside function. This family of functions is displayed for different values
of in Figure 4(a). Note that when , where is the Dirac distribution centered at the origin. In this asymptotic regime, the
convolution filter and its iterates are equal to the identity.

We also define the filtered correlation of the inputs by

where is the kth convolution of with itself and . This is the correlation matrix of the inputs filtered by two different functions.
It is easy to show that this is similar to computing the cross-correlation of the
inputs with the inputs filtered by another function,

(14)

which motivates the definition of the -temporal profile , where . This notation is deliberately similar to that of the transpose operator we use in
the proofs. These functions are shown in Figure 1. We have not found a way to make them explicit; therefore, the following remarks
are simply based on numerical illustrations. When , the temporal profiles are centered. The larger the difference , the larger the center of the bell. The larger the sum , the larger the standard deviation. This motivates the idea that can be thought of as the lagged correlation of the inputs. One can also say that is more blurred than in the sense that the inputs are temporally integrated over a ‘wider’ window in the
first case.

Fig. 1. This shows the -temporal profiles with , i.e., the functions for and k ranging from 0 to 6. For , the temporal profile is even and this also occurs to be true for any . When , the function reaches its maximum for strictly positive values that grow with the
difference . Besides, the temporal profiles are flattened when increases.

Observe that . Therefore, . Thanks to Young’s inequality for convolutions, which says that , it can be proved that .

We intend to express the correlation term as an infinite converging sum involving
these filtered correlations. In this perspective, we use a result we have proved in
[25] to write the solution of a general class of non-autonomous linear systems (e.g., ) as an infinite sum, in the case .

Lemma 3.2Ifis the solution, with zero as initial condition, ofit can be written by the sum below which converges ifWis infor.

where.

Proof See Lemma B.2 in Appendix B.2. □

This is a decomposition of the solution of a linear differential system on the basis
of operators where the spatial and temporal parts are decoupled. This important step
in a detailed study of the averaged equation cannot be achieved easily in models with
non-linear activity. Everything is now set up to introduce the explicit expansion
of the correlation we are using in what follows. Indeed, we use the previous result
to rewrite the correlation term as follows.

Property 3.3The correlation term can be written

Proof See Theorem B.3 in Appendix B.2. □

This infinite sum of convolved filters is reminiscent of a property of Hawkes processes
that have a linear input-output gain [26].

The speed of inputs characterized by μ only appears in the temporal profiles . In particular, if the inputs are much slower than neuronal activity time-scale,
i.e., , then and . Therefore, and the sums in the formula of Property 3.3 are separable, leading to , which corresponds to the heuristic result previously explained.

Therefore, the averaged equation can be explicitly rewritten

(15)

In Figure 2, we illustrate this result by comparing, for different (i.e., we choose in this example), the stochastic system and its averaged version. The above decomposition
has been used as the basis for numerical computation of trajectories of the averaged
system.

Fig. 2. The first two figures, (a) and (b), represent the evolution of the connectivity for original stochastic system (12),
superimposed with averaged system (13), for two different values of ϵ: respectively and , where we have chosen . Each color corresponds to the weight of an edge in a network made of neurons. As expected, it seems that the smaller ϵ, the better the approximation. This can be seen in the picture (c) where we have plotted the precision on the y-axis and ϵ on the x-axis. The parameters used here are , , , . The inputs have a random (but frozen) spatial structure and evolve according to
a sinusoidal function.

3.2.3 Global stability of the equilibrium point

Now that we have found an explicit formulation for the averaged system, it is natural
to study its dynamics. Actually, we prove in the following that if the connectivity
W is kept smaller than , i.e., Assumption 3.1 is verified with , then the dynamics is trivial: the system converges to a single equilibrium point.
Indeed, under the previous assumption, the system can be written , where F is a contraction operator on . Therefore, one can prove the uniqueness of the fixed point with the Banach fixed
point argument and exhibit an energy function for the system.

Theorem 3.4If Assumption 3.1 is verified for, then there is a unique equilibrium point in the invariant subsetwhich is globally, asymptotically stable.

Proof See Theorem B.4 in Appendix B.2. □

The fact that the equilibrium point is unique means that the ‘knowledge’ of the network
about its environment (corresponding by hypothesis to the connectivity) eventually
is unique. For a given input and any initial condition, the network can only converge
to the same ‘knowledge’ or ‘understanding’ of this input.

3.2.4 Explicit expansion of the equilibrium point

When the network is weakly connected, the high-order terms in expansion (15) may be
neglected. In this section, we follow this idea and find an explicit expansion for
the equilibrium connectivity where the strength of the connectivity is the small parameter
enabling the expansion. The weaker the connectivity, the more terms can be neglected
in the expansion.

In fact, it is not natural to speak about a weakly connected learning network since
the connectivity is a variable. However, we are able to identify a weak connectivity index which controls the strength of the connectivity. We say the connectivity is weak
when it is negligible compared to the intrinsic leak term, i.e., is small. We show in the Appendix that this weak connectivity index depends only
on the parameters of the network and can be written

In the asymptotic regime , we have . This index is the ‘small’ parameter needed to perform the expansion. We also define
, which has information about the way is converging to zero. In fact, it is the ratio of the two terms of .

With these, we can prove that the equilibrium connectivity has the following asymptotic expansion in .

Theorem 3.5

Proof See Theorem B.5 in Appendix B.2. □

At the first order, the final connectivity is , the filtered correlation of the inputs convolved with a bell-shaped centered temporal
profile. In the case of Figure 3, this is quite a good approximation of the final connectivity.

Fig. 3. (a) shows the temporal evolution of the input to a neurons network. It is made of two spatially random patterns that are shown alternatively.
(b) shows the correlation matrix of the inputs. The off-diagonal terms are null because
the two patterns are spatially orthogonal. (c), (d), and (e) represent the first order of Theorem 3.5 expansion for different μ. Actually, this approximation is quite good since the percentage of error between
the averaged system and the first order, computed by , have an order of magnitude of 10−4% for the three figures. These figures make it possible to observe the role of μ. If μ is small, i.e., the inputs are slow, then the transient can be neglected and the learned connectivity
is roughly the correlation of the inputs; see (a). If μ increases, i.e., the inputs are faster, then the connectivity starts to encode a link between the
two patterns that were flashed circularly and elicited responses that did not fade
away when the other pattern appeared. The temporal structure of the inputs is also
learned when μ is large. The parameters used in this figure are , , , .

Not only the spatial correlation is encoded in the weights, but there is also some
information about the temporal correlation, i.e., two successive but orthogonal events occurring in the inputs will be wired in the
connectivity although they do not appear in the spatial correlations; see Figure 3 for an example.

3.3 Trace learning: band-pass filter effect

In this section, we study an improvement of the learning model by adding a certain
form of history dependence in the system and explain the way it changes the results
of the previous section. Given that Theorem 2.2 only applies to an instantaneous process,
we will only be able to treat the history-dependent systems which can be reformulated
as instantaneous processes. Actually, this class of systems contains models which
are biologically more relevant than the previous model and which will exhibit interesting
additional functional behaviors. In particular, this covers the following features:

• Trace learning.

It is likely that a biological learning rule will integrate the activity over a short
time. As Földiàk suggested in [27], it makes sense to consider the learning equation as being

where ∗ is the convolution and . Rolls and Deco numerically show [15] that the temporal convolution, leading to a spatio-temporal learning, makes it possible
to perform invariant object recognition. Besides, trace learning appears to be the
symmetric part of the biological STDP rule that we detail in Section 3.4.

• Damped oscillatory neurons.

Many neurons have an oscillatory behavior. Although we cannot take this into account
in a linear model, we can model a neuron by a damped oscillator, which also introduces
a new important time-scale in the system. Adding adaptation to neuronal dynamics is
an elementary way to implement this idea. This corresponds to modeling a single neuron
without inputs by the equivalent formulations

• Dynamic synapses.

The electro-chemical process of synaptic communication is very complicated and non-linear.
Yet, one of the features of synaptic communication we can take into account in a linear
model is the shape of the post-synaptic potentials. In this section, we consider that
each synapse is a linear filter whose finite impulse response (i.e., the post-synaptic potential) has the shape . This is a common assumption which, for instance, is at the basis of traditional
rate based models; see Chapter 11 of [7].

For mathematical tractability, we assume in the following that such that , i.e., the time-scales of the neurons, those of the synapses and those of the learning
windows are the same. Actually, there is a large variety of temporal scales of neurons,
synapses, and learning windows, which makes this assumption not absurd. Besides, in
many brain areas, examples of these time constants are in the same range (≃10 ms).
Yet, investigating the impact of breaking this assumption would be necessary to model
better biological networks. This leads to the following system:

(16)

where the notations are the same as in Section 3.2. The behavior of a single neuron
will be oscillatory damped if is a pure imaginary number, i.e., . This is the regime on which we focus. Actually, the Hebbian linear case of Section 3.2
corresponds to in this delayed system.

To comply with the hypotheses of Theorem 2.2 (i.e., no dependence of the history of the process), we can add a variable z to the system which takes care of integrating the variable v over an exponential window. It leads to the equivalent system (in the limit )

This trick makes it possible to deal with some history-based processes where the
dependence on the past is exponential.

It turns out most of the results of Section 3.2 remain true for system (16) as detailed
in the following. The existence of the solution on is proved in Theorem B.6. The computations show that in the averaged system, the
noise term remains identical, whereas the correlation term is to be replaced by . Besides, Lemma 3.2 can be extended to our delayed system by changing only the temporal
filters; see Lemma 34. Together with Lemma C.3, this proves the result of Theorem B.8.

where

where . Observe that applying Young’s inequality to convolutions leads to . Actually, Lemma C.3 shows that , where is the Bessel function of the first kind. The value of the L1 norm of v is computed in Appendix C.3. It leads to if Δ is a pure imaginary number and else.

Therefore, the averaged system can be rewritten

As before, the existence and uniqueness of a globally attractive equilibrium point
is guaranteed if Assumption 3.1 is verified for ; see Theorem B.9.

Besides, the weakly connected expansion of the equilibrium point we did in Section 3.2.4
can be derived in this case (see Theorem B.10). At the first order, this leads to
the equilibrium connectivity

The second order is given in Theorem B.10. The main difference with the Hebbian linear
case is the shape of the temporal filters. Actually, the temporal filters have an
oscillatory damped behavior if Δ is purely imaginary. These two cases are compared
in Figure 4.

Fig. 4. These represent the temporal filter for different parameters. (a) When , we are in the Hebbian linear case of Appendix B.2. The temporal filters are just
decaying exponentials which averaged the signal over a past window. (b) When the dynamics of the neurons and synapse are oscillatory damped, some oscillations
appear in the temporal filters. The number of oscillations depends on Δ. If Δ is real,
then there are no oscillations as in the previous case. However, when Δ becomes a
pure imaginary number, it creates a few oscillations which are even more numerous
if increases.

These oscillatory damped filters have the effect of amplifying a particular frequency
of the input signal. As shown in Figure 5, if Δ is a pure imaginary number, then is the cross-correlation of the band-pass filtered inputs with themselves. This band-pass
filter effect can also be observed in the higher-order terms of the weakly connected
expansion. This suggests that the biophysical oscillatory behavior of neurons and
synapses leads to selecting the corresponding frequency of the inputs and performing
the same computation as for the Hebbian linear case of the previous section: computing
the correlation of the (filtered) inputs.

Fig. 5. This is the spectral profile for and , where denotes the Fourier transform of . When , the filter reaches its maximum for the null frequency, but if l increases beyond , the filter becomes a band-pass filter with long tails in .

3.4 Asymmetric ‘STDP’ learning with correlated noise

Here, we extend the results to temporally asymmetric learning rules and spatially
correlated noise. We consider a learning rule that is similar to the spike-timing-dependent
plasticity (STDP) which is closer to biological experiments than the previous Hebbian
rules. It has been observed that the strength of the connection between two neurons
depends mainly on the difference between the time of the spikes emitted by each neuron
as shown in Figure 6; see [12].

Fig. 6. This figure represents the synapse strength modification when the post-synaptic neuron
emits a spike. The y-axis corresponds to an additive or multiplicative update of the connectivity. For
instance, in [28], this is for the negative part of the curve. However, we assume an additive update in this
paper. The x-axis is the time at which a pre-synaptic spike reaches the synapse, relatively to
the time of post-synaptic time chosen to be 0.

Assuming that the decay time of the positive and negative parts of Figure 6 are equal, we approximate this function by , where . Actually, this corresponds to . If the neuron has a spiking behavior, then the term is significant when the post-synaptic neuron i is spiking at time t, and then it counts the number of previous spikes from the pre-synaptic neuron j that might have caused the post-synaptic spike. This calculus is weighted by an exponentially
decaying function. This accounts for the left part of Figure 6. The last term takes the opposite perspective. It is significant when the pre-synaptic neuron j is spiking and counts the number of previous spikes from the post-synaptic neuron
i that are not likely to have been caused by the pre-synaptic neuron. The computation
is also weighted by the mirrored function of an exponentially decaying function. This
accounts for the right part of Figure 6. This leads to the coupled system

(17)

where the non-linear intrinsic dynamics of the neurons is represented by f. Indeed, the term is negligible when the neuron is quiet and maximal at the top of the spikes emitted
by neuron i. Therefore, it records the value of the pre-synaptic membrane potential weighted
by the function when the post-synaptic neuron spikes. This accounts for the positive part of Figure 6. Similarly, the negative part corresponds to .

Actually, this formulation is valid for any non-linear activity with correlated noise.
However, studying the role of STDP in spiking networks is beyond the scope of this
paper since we are only able to have explicit results for models with linear activity.
Therefore, we will assume that the activity is linear while keeping the learning rule
as it was derived in the spiking case, i.e., we assume in the system above.

We also use the trick of adding additional variables to get rid of the history-dependency.
This reads

In this framework, the method exposed in Section 3.2 holds with small changes. First,
the well-posedness assumption becomes

Assumption 3.2 There exists such that

where is the maximal eigenvalue of .

Under this assumption, the system is asymptotically well posed in probability (Theorem B.11).
And we show the averaged system is

(18)

where we have used Theorem B.12 to expand the correlation term. The noise term Q is equal to , where is the unique solution of the Lyapunov equation . Lemma D.1 gives a solution for this equation which leads to . In equation (18), the correlation matrices are given by

According to Theorem B.13, the system is also globally asymptotically convergent to
a single equilibrium, which we study in the following.

We perform a weakly connected expansion of the equilibrium connectivity of system
(18). As shown in Theorem B.14, the first order of the expansion is

Writing , where S is symmetric and A is skew-symmetric, leads to

According to Lemma C.1, the symmetric part is very similar to the trace learning
case in Section 3.3. Applying Lemma C.2 leads to

(19)

Therefore, the STDP learning rule simply adds an antisymmetric part to the final
connectivity keeping the symmetric part as the Hebbian case. Besides, the antisymmetric
part corresponds to computing the cross-correlation of the inputs with its derivative.
For high-order terms, this remains true although the temporal profiles are different
from the first order. These results are in line with previous works underlying the
similarity between STDP learning and differential Hebbian learning, where ; see [29].

Figure 7 shows an example of purely antisymmetric STDP learning, i.e., . The final connectivity matrix is therefore antisymmetric as shown in Figure 7(b) and the noise has no impact on learning. It shows the network finally approximates
the connectivity given in (19).

Fig. 7. Antisymmetric STDP learning for a network of neurons. (a) Temporal evolution of the inputs to the network. The three neurons are successively
and periodically excited. The red color corresponds to an excitation of 1 and the blue to no excitation. (b) Equilibrium connectivity. The matrix is antisymmetric and shows that neurons excite
one of their neighbors and are inhibited by the other. (c) Temporal evolution of the connectivity strength. The colors correspond to those
of (b). The connectivity of system (17) corresponds to the plain thin oscillatory curves. The connectivity of the averaged system (18) (with ) corresponds to the plain thick lines. Note that each curve corresponds to the superposition of three connections which
remain equal through learning. The dashed curves correspond to the antisymmetric part in (19). The parameters chosen for this simulation
were , , , , , , , . The system was simulated on the fast time-scale during time steps of size .

4 Discussion

We have applied temporal averaging methods on slow/fast systems modeling the learning
mechanisms occurring in linear stochastic neural networks. When we make sure the connectivity
remains small, the dynamics of the averaged system appears to be simple: the connectivity
always converges to a unique equilibrium point. Then, we performed a weakly connected
expansion of this final connectivity whose terms are combinations of the noise covariance
and the lagged correlations of the inputs: the first-order term is simply the sum
of the noise covariance and the correlation of the inputs.

• As opposed to the former input/ouput vision of the neurons, we have considered the
membrane potential v to be the solution of a dynamical system. The consequence of this modeling choice
is that not only the spatial correlations, but also the temporal correlations are
learned. Due to the fact we take the transients into account, the activity never converges
but it lives between the representation of the inputs. Therefore, it links concepts
together.

The parameter μ is the ratio of the time-scales between the inputs and the activity variable. If
, the inputs are infinitely slow and the activity variable has enough time to converge
towards its equilibrium point. When μ grows, the dynamics becomes more and more transient, it has no time to converge.
Therefore, if the inputs are extremely slow, the network only learns the spatial correlation
of the inputs. If the inputs are fast, it also learns the temporal correlations. This
is illustrated in Figure 3.

This suggests that learning associations between concepts, for instance, learning
words in two different languages, may be optimized by presenting two words to be associated
circularly with a certain frequency. Indeed, increasing the frequency (with a fixed
duration of exposition to each word) amounts to increasing μ. Therefore, the network learns better the temporal correlations of the inputs and
thus strengthens the link between these two concepts.

• According to the model of resonator neuron [30], Section 3.3 suggests that neurons and synapses with a preferred frequency of oscillation
will preferably extract the correlation of the inputs filtered by a band pass filter
centered on the intrinsic frequency of the neurons.

Actually, it has been observed that the auditory cortex is tonotopically organized,
i.e., the neurons are arranged by frequency [31]. It is traditionally thought that this is achieved thanks to a particular connectivity
between the neurons. We exhibit here another mechanism to select this frequency which
is solely based on the parameters of the neurons: a network with a lot of different
neurons whose intrinsic frequencies are uniformly spread is likely to perform a Fourier-like
operation: decomposing the signal by frequency.

In particular, this emphasizes the fact that the network does not treat space and
time similarly. Roughly speaking, associating several pictures and associating several
sounds are therefore two different tasks which involve different mechanisms.

• In this paper, the original hierarchy of the network has been neglected: the network
is made of neurons which receive external inputs. A natural way to include a hierarchical
structure (with layers for instance), without changing the setup of the paper, is
therefore to remove the external input to some neurons. However, according to Theorem 3.5
(and its extensions Theorems B.10 and B.14), one can see that these neurons will be
disconnected from the others at the first order (if the noise is spatially uncorrelated).
Linear activities imply that the high level neurons disconnect from others, which
is a problem. In fact, one can observe that the second-order term in Theorem 3.5 is
not null if the noise matrix Σ is not diagonal. In fact, this is the noise between neurons which will recruit the
high level neurons to build connections from and to them.

It is likely that a significant part of noise in the brain is locally induced, e.g., local perturbations due to blood vessels or local chemical signals. In a way, the
neurons close to each other share their noise and it seems reasonable to choose the
matrix Σ so that it reflects the biological proximity between neurons. In fact, Σ specifies the original structure of the network and makes it possible for close-by
neurons to recruit each other.

Another idea to address hierarchy in networks would be to replace the synaptic decay
term by another homeostatic term [32] which would enforce the emergence of a strong hierarchical structure.

• It is also interesting to observe that most of the noise contribution to the equilibrium
connectivity for STDP learning (see Theorem B.14) vanishes if the learning is purely
skew-symmetric, i.e., . In fact, it is only the symmetric part of learning, i.e., the Hebbian mechanism, that writes the noise in the connectivity.

• We have shown that there is a natural analogous STDP learning for spiking neurons
in our case of linear neurons. This asymmetric rule converges to a final connectivity
which can be decomposed into symmetric and skew-symmetric parts. The first one is
similar to the symmetric Hebbian learning case, emphasizing that the STDP is nothing
more than an asymmetric Hebbian-like learning rule. The skew-symmetric part of the
final connectivity is the cross-correlation between the inputs and their derivatives.

This has an interesting signification when looking at the spontaneous activity of
the network post-learning. In fact, if we assume that the inputs are generated by
an autonomous system , then according to the bottom equation in formula (19), the spontaneous activity
is governed by

In a way, the noise terms generate random patterns which tend to be forgotten by
the network due to the leak term . The only drift is due to which is the expectation of the vector field defining the dynamics of inputs with
a measure being the scalar product between the activity variable and the inputs. In
other words, if the activity is close to the inputs at a given time , i.e., is large, then the activity will evolve in the same direction as what this input
would have done. The network has modeled the temporal structure of the inputs. The
spontaneous activity predicts and replays the inputs the network has learned.

There are still numerous challenges to carry on in this direction.

First, it seems natural to look for an application of these mathematical methods to
more realistic models. The two main limitations of the class of models we study in
Section 3 are (i) the activity variable is governed by a linear equation and (ii) all
the neurons are assumed to be identical. The mathematical analysis in this paper was
made possible by the assumption that the neural network has a linear dynamics, which
does not reflect the intrinsic non-linear behavior of the neurons. However, the cornerstone
of the application of temporal averaging methods to a learning neural network, namely
Property 3.3, is similar to the behavior of Poisson processes [26] which has useful applications for learning neural networks [19,20]. This suggests that the dynamics studied in this paper might be quite similar to
some non-linear network models. Studying more rigorously the extension of the present
theory to non-linear and heterogeneous models is the next step toward a better modeling
of biologically plausible neural networks.

Second, we have shown that the equilibrium connectivity was made of a symmetric and
antisymmetric term. In terms of statistical analysis of data sets, the symmetric part
corresponds to classical correlation matrices. However, the antisymmetric part suggests
a way to improve the purely correlation-based approach used in many statistical analyses
(e.g., PCA) toward a causality-oriented framework which might be better suited to deal
with dynamical data.

Appendix A: Stochastic and periodic averaging

A.1 Long-time behavior of inhomogeneous Markov processes

In order to construct the averaged vector field in the time-scale matching case (), one needs to understand properly the long-time behavior of the rescaled inhomogeneous
frozen process

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Under regularity and dissipativity conditions, [5] proves the following general result about the asymptotic behavior of the solution
of

where and are τ-periodic.

The first point of the following theorem gives the definition of evolution systems of measures, which generalizes the notion of invariant measures in the case of inhomogeneous
Markov processes. The exponential estimate of 2. in the following theorem is a key
point to prove the averaging principle of Theorem 2.2.

1. There exists a uniqueτ-periodic family of probability measuressuch that for all functionsϕcontinuous and bounded,

Such a family is called evolution systems of measures.

2. Furthermore, under stronger dissipativity condition, the convergence of the law ofXtoμis exponentially fast. More precisely, for any, there existandsuch that for allϕin the space ofp-integrable functions with respect to, ,

A.2 Proof of Property 2.3

Property A.2If there exists a smooth subset ℰ ofsuch that

1. The functionsF, G, Σsatisfy Assumptions 2.1-2.3 restricted on.

2. ℰ is invariant under the flow of, as defined in (7).

Then for any initial condition, system (4) is asymptotically well posed in probability andsatisfies the conclusion of Theorem 2.2.

Proof The idea of the proof is to truncate the original system, replacing G by a smooth truncation which coincides with G on ℰ and which is close to 0 outside ℰ. More precisely, for , we introduce a regular function (locally Lipschitz) such that if or and if . We define

Then, we introduce the solution of the auxiliary system

with the same initial condition as .

Let be three positive reals. Let us introduce a few more notations. We will need to consider
a subset of ℰ defined by

We also introduce the following stopping times:

Finally, we define and .

Let us remark at this point that in order to prove that (which is our aim), it is sufficient to work on the bounded stopping time , since . In other words, the realizations of which stay longer than T inside ℰ are not problematic. Therefore, we introduce .

Our first claim is that on finite time intervals , is a good approximation of inside ℰ as long as one chooses β sufficiently small. To prove our claim, we proceed in two steps, first working inside
and then in :

1. For any , one controls the difference between and on since one controls the difference between the drifts. By an application of Lemma A.3
below (we need here the moment Assumption 2.3(i)), there exists a constant C (which may depend on ) such that

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We conclude by an application of the Markov inequality, implying

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2. One needs now to control the situation outside , that is, on . The idea is that while one does not control the difference between G and anymore, one can still choose β sufficiently small such that becomes arbitrary close to ℰ, hence implying that and are arbitrary close with high probability, namely

(23)

With and , one obtains that for sufficiently small β,

(24)

Let us denote . Then, one can split the calculus of according to the event :

where we have used the Cauchy-Schwarz inequality and the moment Assumption 2.3(ii)
(yielding the constant ) in the second line.

So, we deduce by the Markov inequality that is arbitrary small in probability.

From the combination of 1. and 2., we deduce that one can choose β small enough such that

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We can now proceed to the application of Theorem 2.2 to the truncated system. As
remains in , one can extend smoothly F and Σ outside ℰ so that satisfies Assumptions 2.1-2.2. Therefore, one can apply Theorem 2.2 to the auxiliary
system: for all ,

where w is defined by (8). As a consequence, there exists such that for all ϵ with ,

Then, as , one deduces that for all ϵ with ,

that is to say,

We know by assumption 2. of the statement of Property 2.3, for all , , so we conclude the proof by observing that for all ,

□

In the following lemma, we show that the solutions of two SDEs, whose drifts are close
on a subset of the state space, remain close on a finite time interval. The difficulty
here lies in the fact that we deal with only locally Lipschitz coefficients.

2. Local Lipschitz assumption: for allwith, there exists a constantsuch that

3. Boundedness assumption: there existsandsuch that

and if, then there existssuch that.

Under the above assumptions, there exists a constantC (depending on the quantities defined above, but not onξ) such that

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Proof Although the Lipschitz constant is not bounded on ℋ, we can use the boundedness assumption
to show that the probability of reaching a level R before time T will be very small for large R, and then use the classical strategy inside where everything works as if the coefficients were globally Lipschitz. A similar
strategy is used in [33] to prove a strong convergence theorem for the Euler scheme without the global Lipschitz
assumption. We adapt here the ideas of their proof to our setting.

Therefore, we introduce the following stopping times:

We also denote .

Splitting the following expectation according to the value of ρ, and applying the Young inequality,

we obtain, for any ,

Then we use the boundedness assumption and the Markov inequality to deduce that

Now, we can focus on the supremum of the error before time ρ. We first apply the Cauchy-Schwarz inequality

Then, we use the local Lipschitz and the boundedness assumptions, together with the
Doob inequality (the first inequality) to deal with the stochastic integral: for any
,

We then apply the Gronwall lemma

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Finally, we choose d small enough such that

and R large enough such that

yielding

□

Appendix B: Proofs of Section 3

B.1 Notations and definitions

Throughout the paper, lower-case normal letters are constants, lower-case bold letters
are vectors or vector-valued functions, and upper-case bold letters are matrices.

• are parameters of the network. We also define for Section 3.3 and , a fixed noise matrix, for Section 3.4. We write .

• is the number of neurons in the network.

• is the field of membrane potential in the network.

• is the field of inputs to the network. We write

• is the tensor product between u and v, which simply means .

• is the connectivity of the network. Throughout the paper, we assume .

• is the scalar product between two vectors .

• for is the norm of , i.e., . And similarly for the connectivity matrices of with a double sum.

• .

• is the transpose of the matrix .

• is the cross-correlation matrix of two compactly supported and differentiable functions
from ℝ to , i.e.,

• H is the Heaviside function, i.e.,

• The real functions

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are integrable on ℝ.

B.1.1 Notations for the Appendix

The computations involve a lot of convolutions and, for readability of the Appendix,
we introduce some new notations. Indeed, we rewrite the time-convolution between u and g, an integrable function on ℝ,

This suggests one should think of v as a semi-continuous matrix of and of as a continuous matrix of , such that and . Indeed, in this framework the convolution with g is nothing but the continuous matrix multiplication between v and a continuous Toeplitz matrix generated row by row by g. Hence, the operator ‘⋅’ can be though of as a matrix multiplication.

Therefore, it is natural to define , where is the transpose of , i.e., the continuous Toeplitz matrix generated row by row by and . Thus, for g and h, two integrable functions on ℝ, we can rewrite

where and ℋ are their associated continuous matrices. More generally, the bold curved letters
, , represent these continuous Toeplitz matrices which are well defined through their
action as convolution operators with g, v, and w. The previous formulation naturally expresses the symmetry of relation (14).

B.2 Hebbian learning with linear activity

In this part, we consider system (12).

B.2.1 Application of temporal averaging theory

Theorem B.1If Assumption 3.1 is verified for, then system (12) is asymptotically well posed in probability and the connectivity matrix, the solution of system (12), converges toW, in the sense that for all,

whereWis the deterministic solution of

whereis the-periodic attractor of, whereis supposed to be fixed.

Proof We are going to apply Property 2.3. For , consider the space

First, since is strictly positive for W in , Assumptions 2.1-2.2 are satisfied on . Then, we only need to compute the averaged vector field and show that is invariant under the flow of .

1. Computation of the averaged vector field :

The fast variable is linear, the averaged vector field is given by (10). This reads

where is the probability density function of the Gaussian law with mean v and covariance Q. And Q is the unique solution of (9), with . This leads to .

Therefore,

The integral term in the equation above is the correlation matrix of the -periodic function . To rewrite this term, we define such that . can be seen as a matrix gathering the history of , i.e., each column of corresponds to the vector for a given . It turns out

Therefore,

According to the results in Section 2, the solutions of a differential system with
such a right-hand side are close to that of the initial system (12). Hence, we focus
exclusively on it and try to unveil the properties of its solutions which will be
retrospectively extended to those of the initial system (12).

2. Invariance of under the flow of (13):

Here we assume that and we want to prove that the trajectory of W is in , too.

(a) Symmetry:

It is clear that each term in is symmetric. Their sum is therefore symmetric and so is .

(b) Inequality :

The correlation term is a Gramian matrix and is therefore positive. Because is assumed to be positive, therefore, its inverse is also positive. Therefore, if
is an eigenvector of associated with a null eigenvalue, then . Thus, the trajectories of (13) remain positive.

(c) Inequality :

The argument here is that of the inward pointing subspace. We intend to find a condition
under which the flow is pointing inward the space . Roughly speaking, this will be done by defining a real valued function g strictly negative on the subspace and positive outside and then showing that its
gradient (or differential) on the border goes in the opposite direction of the flow,
i.e., for .

For all such that , define a family of positive numbers whose supremum is written and a family of functions such that

Observe that the differential of at W applied to J is . For , i.e., , compute

Therefore, for

where

(31)

Now write with . Equation (31) becomes

When there exists p such that (which corresponds to Assumption 3.1), then their exists a ball of radius pl on which the dynamics is pointing inward. It means any matrix W whose maximal eigenvalue is will see this eigenvalue (and those which are sufficiently close to it, i.e., for which is sufficiently small) decreasing along the trajectories of the system. Therefore,
the space is invariant by the flow of the system iff Assumption 3.1 is satisfied.

• Upper bound of A:

Applying Cauchy-Schwarz leads to

However, for

Therefore, .

• Upper bound of B:

Observe that for J a positive definite matrix whose eigenvalues are the , then the spectrum of is . Therefore, . Therefore, if , then .

Using the previous observation and Cauchy-Schwarz leads to

The trajectories of system (13) with the initial condition in are defined on and remain bounded. Indeed, if , the connectivity will stay in , in particular along the trajectories, more precisely is a strictly positive constant since . Because is also bounded by , is bounded. The right-hand side of system (13) is the sum a bounded term and a linear
term multiplied by a negative constant; therefore, the system remains bounded and
it cannot explode in finite time: it is defined on . □

B.2.2 An expansion for the correlation term

We first state a useful lemma.

Lemma B.2Ifis the solution, with zero as initial condition, of, it can be written by the sum below which converges ifWis infor.

where.

Proof It can be proven as a trivial rewriting of the variation of parameters formula for
linear systems. A more general approach, which extends to delayed systems, was developed
by Galtier and Touboul [25]; see the first example for the proof of this lemma. □

This is useful to find the next result.

Property B.3The correlation term can be written

Proof We can use Lemma 3.2 with and compute the cross product .

Therefore, consider instead of u. A change of variable shows that . Therefore,

□

B.2.3 Global stability of the single equilibrium point

Theorem B.4If Assumption 3.1 is verified for, then there is a unique equilibrium point in the invariant subsetwhich is globally, asymptotically stable.

Proof For this proof, define .

First, we compute the differential of F and show it is a bounded operator. Second, we show it implies the existence and uniqueness
of an equilibrium point under some condition. Then, we find an energy for the system
which says the fixed point is a global attractor. Finally, we show the stability condition
is the same as Assumption 3.1 for .

1. We compute the differential of each term in F: The differential of F at W is the sum of these two terms.

• Formally write the second term . To find its differential, compute and keep the terms at the first order in J. Before computing the whole sum, observe that

This leads to

• Write . We can write and use the chain rule to compute the differential of Q at W, which gives . Therefore,

2. We want to compute the norm of for . First, observe that for three square matrices A, B, and C,

for the vectors of the canonical basis of . This leads to . Therefore, because ,

Therefore,

This inequality is true for all J with ; therefore, it is also true for the operator norm

Therefore, F is a k-Lipschitz operator where . This means .

3. The equilibrium points of system (15) necessarily verify the equation . If

(32)

then is a contraction map from to itself. Therefore, the Banach fixed point theorem says that there is a unique
fixed point which we write .

4. We now show that, under assumption (32), is an energy function for the system (which is a rescaled version of system (15)).

Indeed, compute the derivative of this energy along the trajectories of the system

The energy is lower-bounded, takes its minimum for and the decreases along the trajectories of the system. Therefore, is globally asymptotically stable if assumption (32) is verified.

5. Observe that if Assumption 3.1 is verified for , then . Therefore, Assumption 3.1 implies that (32) is also true. This concludes the proof. □

B.2.4 Explicit expansion of the equilibrium point

Recall the notations and .

Theorem B.5

Actually, it is possible to compute recursively the nth term of the expansion above, although their complexity explodes.

Proof Define the smallest value in such that Assumption 3.1 is valid. This implies

The weak connectivity index controls the ratio of the connection over the strength of intrinsic dynamics. Indeed,
these two variables are of the same order because

We want to approximate the equilibrium , i.e., the solution of , in the regime . Define such that . We abusively write such that

Recalling leads to

Now, we write a candidate , then we chose the terms so that the first mth orders in vanish. This implies that , where . Then, we use the fact that the minimal absolute value of the eigenvalues of is larger than . Indeed, it means

i.e., .

Thus, we need to find the such that the first mth orders in vanish. Therefore, we need to expand all the terms in . The first term is obvious. In the following, we write the second term associated to the correlations and look for an explicit expression of the such that . Second, we write the third term associated to the noise and look for an explicit expression of the such that .

• Finding the :

First, observe that

(33)

This leads to

The ath term in the power expansion in verifies . More precisely, this reads

This equation is scary but it reduces to simple expressions for small .

• Finding the :

Using equation (33) leads to

The ath term in the power expansion in verifies . More precisely, this reads

Therefore,

Therefore, it is easy to compute for . By definition , which leads to the result. □

B.3 Trace learning with damped oscillators and dynamic synapses

Theorem B.6If Assumption 3.1 is verified for, then system (16) is asymptotically well posed in probability and the connectivity matrix, solution of system (16), converges toWin the sense that for all,

Starting from this system, the structure of the proof of Theorem 3.1 remains unchanged.
The correlation term is to be replaced by . The noise term we are looking for is in the Lyapunov equation (see (9)) below

Because the learning rule is symmetric, then the space of symmetric matrices is invariant
and we can restrict this section to the symmetric case. It is easy to show that this
Lyapunov equation has a unique solution, because the sum of two eigenvalues of the
drift matrix is never null (provided W stays in ). This leads to the system

One solution of equation (a) is . Equation (c) defines properly. Indeed, because W is symmetric, so is and . Similarly, equation (b) defines but it remains to be checked that this definition is that of a symmetric matrix.
In fact, it works because W is assumed symmetric and the noise has no off-diagonal terms. Indeed, in this case,
. This solution is thus the unique solution of the Lyapunov equation.

Therefore,

Thus, this application of Theorem 2.2 to the instantaneous system with , leads to the previous averaged equation. To recover the initial case (16), we can
let . We see that the function tends to

which we will rewrite for simplicity in the following. Thus, this definition of defines the averaged system for the original equation (16).

In the derivation of the condition under which remain smaller than lp, the upper bound of the term A changes as follows. Define so that for all . Because we assume , the variation of parameters formula for linear retarded differential equations with
constant coefficients (see Chapter 6 of [34]) reads where the resolvent U is the solution of . We use Corollary 1.1 of Chapter 6 of [34], which is based on Grönwall’s lemma, to claim that . Therefore,

Then, we used Young’s inequality for convolution to get .

Therefore, the upper bound of A remains unchanged.

Therefore, the polynomial P remains the same and Assumption 3.1 is still relevant to this problem. □

Lemma B.7Ifis the solution, with zero as initial condition, of, it can be written by the sum below which converges ifWis infor.

whereHis the Heaviside function, . If Δ is a pure imaginary number, the expression above still holds with the hyperbolic functionsshandchbeing turned into classical trigonometric functions sin and cos and Δ being replaced by its modulus.

where is the convolution operator generated by (see Appendix C for details). Observe that applying Young’s inequality for convolutions
leads to .

Therefore, we can rewrite Theorem 3.3 into

Theorem B.8

Proof Similar to that of Theorem 3.3. □

Theorem B.9If Assumption 3.1 is verified for, there is a unique equilibrium point which is globally, asymptotically stable.

Proof Similar to the proof of Theorem B.4. □

With the same definitions for and , we can show

Theorem B.10The weakly connected expansion of the equilibrium point is

Proof Define so that

So, the expansion will be in orders of with .

Therefore,

Actually, it is possible to compute recursively the nth terms, although their complexity explodes. Therefore, it is easy to compute for . By definition , which leads to the result. □

B.4 STDP learning with linear neurons and correlated noise

Consider the following n-dimensional stochastic differential system:

where u is a continuous input in , , , and is n-dimensional Brownian noise, and for all , where H is the Heaviside function. Recall the well-posedness Assumption 3.2

Assumption B.1 There exists such that

Theorem B.11If Assumption 3.2 is verified for, then system (17) is asymptotically well posed in probability and the connectivity matrix, the solution of system (17), converges toWin the sense that for all,

Proof We recall the instantaneous reformulation of the original system (17)

With this linear expression, the structure of the proof of Theorem 3.1 remains unchanged.
The correlation term is to be replaced by . The noise term we are looking for is in the Lyapunov equation (see (9)) below

This leads to the system

(34)

The matrix is the solution of a Lyapunov equation (see equation (a)). Lemma D.1 gives an explicit
solution: . Equation (b) leads to

We see that it does not depend on , which, once Theorem 2.2 is applied, can be considered null so that the average system
corresponds to the original system (17).

Therefore,

(35)

We show that for W already in , it will stay forever in :

1. Inequality :

Decomposing the connectivity as leads to . By hermiticity of S and A, and are real numbers. This means we only have to show that the eigenvalues of S remain positive along the dynamics. Taking the symmetric part of equation (35) leads
to

Suppose we take an initial condition . It is clear that if and are always positive, then S will remain positive. This would prove the result. Therefore, focus on

• Proving :

According to the first point of Lemma C.1, . Therefore, is a Gramian matrix and therefore it is positive.

• Proving :

is the covariance matrix of the random value z, therefore, it is positive-semi-definite.

2. Inequality :

For all such that , define a family of positive numbers whose supremum is written and a family of functions such that

Because g is linear, . For , i.e., , compute

• Upper bound of A:

Cauchy-Schwarz leads to

As before, we can use Young’s inequality for convolutions to find an upper bound
of A which reads

• Upper bound of B:

According to Proposition 11.9.3 of [35] the solution of the Lyapunov equation (a) in system (34) can be rewritten

because is not singular due to the fact .

Observe that for A a positive matrix whose eigenvalues are the , then the spectrum of is . Therefore, . Therefore, if , then . This leads to

Then we apply the same arguments to say that

The rest of the proof is identical to the Hebbian case. Assumption 3.1 is changed
to Assumption 3.2 for to be invariant by the flow . □

Define

such that .

In this framework, one can prove

Theorem B.12The correlation term can be written

Proof Similar to that of Theorem 3.3. □

Theorem B.13If Assumption 3.2 is verified for, there is a unique equilibrium point which is globally, asymptotically stable.

Proof Similar to the previous case. □

Now, we proceed as before by defining

Theorem B.14

Proof First, we need to work on the noise term . Recall is the solution of the Lyapunov equation . Lemma D.1 says that

is a well-defined solution. We now use the fact that to show that

and therefore

Thus, writing and , the noise term is

Define such that . We improperly write such that

This leads to

We are looking for and in the expansions and . Recall

Therefore,

Leading to

This equation is scary but it reduces to simple expressions for small .

Recall that to get the result. □

Appendix C: Properties of the convolution operators , , and

Recall , , and are convolution operators respectively generated by , v, and w defined in (30). Their Fourier transforms are respectively

C.1 Algebraic properties

Lemma C.1

Proof Compute

Therefore, if , then , and if , then . The result follows. □

Lemma C.2

whereis the time-differentiation operator, i.e., .

Proof and are two convolution operators respectively generated by and . The Fourier transform of is . Therefore, the Fourier transform of is

Because , taking the inverse Fourier transform of gives the result. □

Lemma C.3

Besides, if, is a convolution operator generated by

whereis the Bessel function of the first kind. If, the formula above holds if one replacesby, the modified Bessel function of the first kind.

Proof We want to compute . Compute the Fourier transform of , where is the result of k convolutions of v with itself

This proves the first result.

Then observe that

The last integral can be analytically computed with the help of Bessel functions.
In fact, it gives different results depending on the nature of Δ (whether it is real
or imaginary).

• If , then defining , the modified Bessel function of the first kind, leads to

• If , then defining , the Bessel function of the first kind, leads to

This concludes the proof. □

C.2 Signed integral

1. .

2. For , compute

3. Similarly,

C.3 L1 norm

• For , i.e., , then

1. and .

2. and .

3. and .

• For , i.e., Δ is a pure imaginary, we rewrite and observe that

1. and .

2. which changes sign on . Therefore,

3. which also changes sign on . We have not found a way to compute and write the result elegantly.

Appendix D: Solution of a Lyapunov equation

Lemma D.1The solution of the following Lyapunov equation

whereis

(36)

Proof First, observe that if and , then . Therefore, X is well defined by equation (36).

Observe that . Assuming X is defined by equation (36), then based on the fact L commutes with any matrix (because it is a scalar matrix),

□

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

GW developed the theory of temporal averaging presented in this paper. MG applied
this theory to learning neural networks and did the numerical simulations. Both authors
read and approved the final manuscript.

Acknowledgements

MG thanks Olivier Faugeras for his support. MG was partially funded by the ERC advanced
grant NerVi nb227747, by the IP project BrainScaleS #269921 and by the région PACA,
France. GW thanks L. Ryzhik from Stanford University, Department of Mathematics for
his hospitality during 2010-2011 where part of this work was achieved.